| Résume | O-minimal structures on the real field have many desirable properties. As examples:
(a) Hausdorff (and even packing) dimension agrees with topological dimension on locally closed definable sets.
(b) Locally closed definable sets have few rational points (in the sense of the Pila-Wilkie Theorem).
(c) For each positive integer p, every closed definable set is the zero set of a definable C^p function.
(d) Connected components of definable sets are definable.
But to what extent is o-minimality necessary for these properties to hold? I will discuss this question, and illustrate via examples as to why one might care about answers. |
